6.5 State and Control Path Constraints

Core idea¶

State and Control Path Constraints must be treated as a system-level decision rather than an isolated technique. For an active suspension with complete road preview and a wave-energy converter with known future waves, state what is fixed, what is optimized, what information is available, and what equations define feasibility.

The relevant quantities are $p$ , $u(t)$ , $x(t)$ , and known $d(t)$ . The chapter-level formulation is

\min_{p,u,x}\Phi+\int L\,dt\;\mathrm{s.t.}\;\dot x=f(t,x,u,p,d).

For this section, trace how the choice changes performance limit, the active constraints, and the implementable engineering design. A method is useful only when its assumptions are explicit and its result answers the same system question as the baseline.

Engineering interpretation¶

Ask three questions:

Which physical, informational, computational, or economic resource changed?
Which objective component or active constraint made the change valuable?
Does the conclusion survive model, disturbance, initialization, uncertainty, and implementation checks?

A practical action is to translate insight to feedback. Record units and assumptions before optimization, report component objectives and margins afterward, and verify the result using an independent calculation or higher-fidelity model.

Activity 6.5: quantitative design audit¶

Activity 6.5: quantitative design audit

An OLOC suspension design is optimized on nodes separated by $h=0.20$ s. Suspension travel must satisfy $|s(t)|\leq0.050$ m and actuator force must satisfy $|u(t)|\leq2.50$ kN for the entire maneuver. The optimizer reports:

Node time (s)	0.00	0.20	0.40	0.60	0.80	1.00
$s$ (m)	0.000	0.038	0.049	0.047	0.030	0.000
$u$ (kN)	0.00	2.40	2.50	-2.45	-2.30	0.00

An independent simulation evaluates the interval midpoints:

Midpoint time (s)	0.10	0.30	0.50	0.70	0.90
$s$ (m)	0.021	0.052	0.054	0.043	0.017
$u$ (kN)	1.70	2.62	2.58	-2.70	-1.50

Compute every node constraint margin and explain why the optimizer declares the trajectory feasible.
Compute midpoint violations in physical units and as percentages of the corresponding limits. Identify the worst state and control violation.
Approximate the integrated violation measures $V_s=h\sum\max(0,|s|-0.050)$ and $V_u=h\sum\max(0,|u|-2.50)$ using the midpoint data.
Determine tightened node limits that would cover the observed maximum node-to-midpoint growth factors. Include an additional 5% validation reserve.
If halving the mesh spacing reduces each midpoint overshoot by a factor of four, predict the new worst travel and force and test feasibility against the original limits.